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A166134 a(n+1) is the smallest divisor of a(n)^2+1 that does not yet appear in the sequence, with a(1) = 1.

%I A166134 #5 Sep 03 2015 18:49:22
%S A166134 1,2,5,13,10,101,5101,26,677,45833,65,2113,446477,130,16901,41,29,421,
%T A166134 17,58,673,45293,25,313,97,941,34057,50,61,1861,1229,773,59753,89,34,
%U A166134 1157,82,269,194,617,38069,55740337,145,10513,11052317,12215371106849
%N A166134 a(n+1) is the smallest divisor of a(n)^2+1 that does not yet appear in the sequence, with a(1) = 1.
%C A166134 All members of the sequence can be represented as the sum of two relatively prime numbers (A008784). It appears that the sequence is infinite and that all such numbers are present.
%H A166134 Ivan Neretin, <a href="/A166134/b166134.txt">Table of n, a(n) for n = 1..1000</a>
%e A166134 After a(4)=13, the divisors of 13^2+1=170 are 1,2, 5, 10, 17, 34, 85, 170. 1, 2, and 5 have already occurred, so a(5) = 10.
%t A166134 Nest[Append[#, Min[Complement[Divisors[#[[-1]]^2 + 1], #]]] &, {1}, 45] (* _Ivan Neretin_, Sep 03 2015 *)
%o A166134 (PARI) invec(v,x,n)=for(i=1,n,if(v[i]==x,return(1)));0
%o A166134 bl(n)={local(v,d,ds);
%o A166134 v=vector(n,i,1);
%o A166134 for(i=2,n,
%o A166134 ds=divisors(v[i-1]^2+1);
%o A166134 for(k=2,#ds,d=ds[k];if(!invec(v,d,i-1),v[i]=d;break)));
%o A166134 v}
%Y A166134 Cf. A166133, A008784, A031439, A002522.
%K A166134 nonn
%O A166134 1,2
%A A166134 _Franklin T. Adams-Watters_, Oct 07 2009